Pendulums Simple pendulums ignore friction, air resistance, mass of string Physical pendulums take into account mass distribution, friction, air resistance The force that pulls the mass back towards equilibrium is the restoring force

Pendulums If the restoring force is proportional to the displacement, then the pendulum’s motion is simple harmonic.

Pendulums For small angles (less than 15°) the pendulum is in simple harmonic motion. Gravitational PE increases as the displacement increases. Pendulums have gravitational PE and springs have elastic PE. For pendulums: x↑, PE g ↑ PE g = 0 at equilibrium PE = max; KE = 0 PE = 0; KE = max PE = max; KE = 0

Pendulums The mechanical energy of a simple pendulum is conserved in a frictionless system. A pendulum’s mechanical energy changes as the pendulum oscillates.

Pendulums Amplitude = the maximum displacement from equilibrium, measured in radians or meters. Period (T) = the time it takes for one complete cycle of motion, measured in seconds. Frequency (f) = the number of cycles or vibrations per unit of time, measured in hertz (Hz). 1 Hz = s -1

Pendulums Period and frequency are inversely proportional: f = 1/T or T = 1/f

Pendulums The period of a simple pendulum depends on pendulum length and free-fall acceleration (on Earth it is 9.81 m/s 2 T = 2π√(L/g) Period = 2π * square root of (length divided by free-fall acceleration)

Pendulums Shorter pendulums have shorter periods when the acceleration due to gravity is the same. Mass does not affect the period because while the heavier mas provides a larger restoring force, it also needs a larger force to achieve the same acceleration. Therefore when acceleration due to gravity is the same, pendulums with bobs of different masses (and same length) will have the same period. Amplitude does not affect the period when the angle is less than 15°.

Springs But for springs, the heavier the mass on the end, the greater the period: T = 2π√(m/k) Period = 2π * square root of (mass divided by spring constant)

Pendulums Ex: You are designing a pendulum clock to have a period of 1.0 s. How long should the pendulum be? G: T = 1.0 sS: 1.0 s = 2π √(L/9.81m/s 2 ) g = 9.81 m/s 2 S: 0.25 m U: L E: T = 2π√(L/g)